During the last few years, metamaterials have received significant attention in research due to their anomalous electromagnetic properties and, hence, their potential for unique applications. See R. A. Shelby et al., Science 292 (5514), 77 (2001). In 1968, Veselago proposed a particular class of metamaterials, referred to as a left-handed metamaterial, which had several unusual properties, phase and energy flux of opposing sign, a negative index of refraction, reversal of the Doppler effect, and flat lens focusing. See V. G. Veselago, Soviet Physics—Uspekhi 10 (4), 509 (1968). More generally, a metamaterial is any artificial material that exhibits electromagnetic properties that are not necessarily displayed by the constituent elements. This is primarily due to resonant effects arising from the periodic orientation of the individual elements, which are typically sub-wavelength in size. Thus, electromagnetic metamaterials can theoretically exhibit any value of permittivity or permeability near the resonance frequency, including negative values. This prospect has led to the proposal of applications ranging from superlenses and the enhancement of antenna systems, to arguably even electromagnetic cloaking. See J. B. Pendry, Physical Review Letters 85 (18), 3966 (2000); K. B. Alici and E. Özbay, Physica Status Solidi B 244 (4), 1192 (2007); and J. B. Pendry et al., Science 312 (5781), 1780 (2006).
Despite the growing body of work involving metamaterials, little consensus has emerged regarding the optimal structure for producing a given set of electromagnetic properties, although a few general design templates such as the split-ring resonator (SRR) have become popular, largely due to their relative ease of fabrication. See J. B. Pendry et al., IEEE Transactions on Microwave Theory and Techniques 47 (11), 2075 (1999). The SRR element can be used to achieve a negative permeability in the vicinity of a magnetic resonance frequency. As shown in FIG. 1, the simplest form of the SRR 10 is a planar metallic ring 11 with a gap 12. The ring 11 has an outer dimension l and a metal linewidth w. The gap 12 has a width g. In essence, the SRR 10 is a small LC circuit consisting of an inductance L and a capacitance C. The ring 11 forms one winding of a coil (the inductance), and the ends at the gap 12 form the plates of a capacitor. Electromagnetic radiation directed into the plane of the SRR (i.e., in the z direction) induces a ring current I in the ring. Metamaterials comprise an array of such subwavelength metallic resonator elements within or on an electrically insulating or semiconducting substrate. Dense packing of SRRs, using lattice constants smaller than the LC resonance wavelength, creates a metamaterial that can exhibit a magnetic and electric resonance at the resonant frequency, ωLC=1/√{square root over (LC)}. Two resonances are observed when exciting the SRR structure shown with incident radiation having polarization perpendicular to the gap (i.e., electric field E parallel to the arm containing the gap, as shown). The LC resonance corresponding to the ring current leads to a magnetic dipole moment perpendicular to the SRR plane and an electric dipole moment parallel to the incident electric field. A shorter wavelength Mie resonance is also excited, corresponding to an electric dipole oscillating in the arm opposite the gap. With incident radiation polarized parallel to the gap, only a Mie resonance corresponding to electric dipoles oscillating in the two arms parallel to the gap is observed. The resonances can be strengthened by adding additional, concentric rings, each ring having a gap, to the simple SRR structure. In principle, the resonator response is scalable from radio to infrared and optical frequencies. For the simple SRR described above, both the inductance and capacitance scale proportionally to SRR size, provided that all SRR dimensions are scaled down simultaneously and that the metal retains a high conductivity. Therefore, the resonant frequency scales inversely with a normalized size. Depending on the size, such SRRs can be fabricated using bulk and micromachining techniques known in the art. See D. R. Smith et al., Phys. Rev. Lett. 84, 4184 (2000); J. B. Pendry et al., Science 312, 1780 (2006); D. R. Smith et al., Science 305, 778 (2004); Xin-long Xu et al., J. Opt. Soc. Am. B. 23 (6), 1174 (2006); M. W. Klein et al., Optics Letters 31 (9), 1259 (2006); and C. Enkrich et al., Phys. Rev. Lett. 95, 203901 (2005).
However, the number of different metamaterial element designs that have been published almost rivals the number of groups investigating metamaterials. The variety of designs is a reflection of the lack of a formalized method for designing such structures. Thus, metamaterial design is often a cyclic process of “educated guesswork” and trial-and-error, making extensive use of numerical simulations that are occasionally combined with optimization techniques such as generic algorithms.
The problem of designing an electromagnetic metamaterial is complicated by the pseudo-infinite parameter space governing such materials. For example, a metamaterial unit cell composed of a simple circular SRR (similar to the character “C”) on the six faces of a cube comprises 46=4096 possible orientations of the cell in the most general case. Even if the orientations that are indistinguishable due to symmetry are eliminated and the quasi-static limit is invoked, the number of possibilities that would have to be tried in a brute-force approach only reduces to 128. Given typical simulation times on the order of tens of hours for a fully-vectorial numerical electromagnetic simulation for this type of structure, the problem quickly becomes intractable. Additionally, such simulations only provide the net result with limited insight into the inter-element interactions.
Therefore, a need remains for a method to design a metamaterial with predictable functionality and tailorable electromagnetic properties.